1 Statistical Tools in Finance and Insurance ˇ ıˇ Pavel Czek, Wolfgang Hărdle, Rafal Weron a November 25, 2003 Contents I Finance Stable distributions in finance 11 Szymon Borak, Wolfgang Hărdle, Rafal Weron a 1.1 Introduction 11 1.2 α-stable distributions 12 1.2.1 Characteristic function representation 14 1.2.2 Simulation of α-stable variables 16 1.2.3 Tail behavior 18 Estimation of parameters 18 1.3.1 Tail exponent estimation 19 1.3.2 Sample Quantiles Methods 22 1.3.3 Sample Characteristic Function Methods 23 Financial applications of α-stable laws 26 1.3 1.4 Tail dependence 33 Rafael Schmidt 2.1 Tail dependence and copulae 33 2.2 Calculating the tail-dependence coefficient 36 Contents 2.2.1 Archimedean copulae 36 2.2.2 Elliptically contoured distributions 37 2.2.3 Other copulae 40 2.3 Estimating the tail-dependence coefficient 43 2.4 Estimation and empirical results 45 Bibliography 53 Implied Trinomial Trees 55 Karel Komor´d a 3.1 Introduction 55 3.2 Basic Option Pricing Overview 57 3.3 Trees and Implied Models 59 3.4 ITT’s and Their Construction 62 3.4.1 Basic insight 62 3.4.2 State space 64 3.4.3 Transition probabilities 66 3.4.4 Possible pitfalls 67 3.4.5 Illustrative examples 68 Computing Implied Trinomial Trees 74 3.5.1 Basic skills 74 3.5.2 Advanced features 81 3.5.3 What is hidden 84 Bibliography 87 3.5 Functional data analysis 89 Michal Benko, Wolfgang Hărdle a 4.1 Introduction 89 Contents 5 Nonparametric Productivity Analysis 91 Wolfgang Hărdle, Seok-Oh Jeong a 5.1 Introduction 91 5.2 Nonparametric Hull Methods 93 5.2.1 An Overview 93 5.2.2 Data Envelopment Analysis 94 5.2.3 Free Disposal Hull 94 5.3 DEA in Practice : Insurance Agencies 95 5.4 FDH in Practice : Manufacturing Industry 96 Money Demand Modelling 103 Noer Azam Achsani, Oliver Holtemăller and Hizir Sofyan o 6.1 Introduction 103 6.2 Money Demand 104 6.2.1 General Remarks and Literature 104 6.2.2 Econometric Specification of Money Demand Functions 105 6.2.3 Estimation of Indonesian Money Demand 108 Fuzzy Model Identification 113 6.3.1 Fuzzy Clustering 113 6.3.2 Takagi-Sugeno Approach 114 6.3.3 Model Identification 115 6.3.4 Modelling Indonesian Money Demand 117 Conclusions 118 Bibliography 121 6.3 6.4 The exact LR test of the scale in the gamma family Milan Stehl´ ık 125 Contents 7.1 Introduction 125 7.2 Computation the exact tests in the XploRe 127 7.3 Illustrative examples 128 7.3.1 Time processing estimation 128 7.3.2 Estimation with missing time-to-failure information 132 7.4 Implementation to the XploRe 137 7.5 Asymptotical optimality 138 7.6 Information and exact testing in the gamma family 139 7.7 The Lambert W function 140 7.8 Oversizing of the asymptotics 141 Bibliography 143 Pricing of catastrophe (CAT) bonds 147 Krzysztof Burnecki, Grzegorz Kukla,David Taylor 8.1 Introduction Extreme value theory 147 149 Krzysztof Jajuga, Daniel Papla 9.1 Introduction 149 10 Applying Heston’s stochastic volatility model to FX options markets151 Uwe Wystup, Rafal Weron 10.1 Introduction 11 Mortgage backed securities: how far from optimality 151 153 Nicolas Gaussel, Julien Tamine 11.1 Introduction 12 Correlated asset risk and option pricing 153 155 Contents Wolfgang Hărdle, Matthias Fengler, Marc Tisserand a 12.1 Introduction II Insurance 13 Loss distributions 155 157 159 Krzysztof Burnecki,Grzegorz Kukla, Rafal Weron 13.1 Introduction 14 Visualization of the risk process 159 161 Pawel Mista, Rafal Weron 14.1 Introduction 15 Approximation of ruin probability 161 163 Krzysztof Burnecki, Pawel Mista, Aleksander Weron 15.1 Introduction 16 Deductibles 163 165 Krzysztof Burnecki, Joanna Nowicka-Zagrajek, Aleksander Weron, A Wyloma´ska n 16.1 Introduction 17 Premium calculation 165 167 Krzysztof Burnecki, Joanna Nowicka-Zagrajek, W Otto 17.1 Introduction 167 18 Premium calculation when independency and normality assumptions are relaxed 169 W Otto 18.1 Introduction 169 Contents 19 Joint decisions on premiums, capital invested in insurance company, rate of return on that capital and reinsurance 171 W Otto 19.1 Introduction 20 Stable Levy motion approximation in collective risk theory 171 173 Hansjoerg Furrer, Zbigniew Michna, Aleksander Weron 20.1 Introduction 21 Diffusion approximations in risk theory 173 175 Zbigniew Michna 21.1 Introduction 175 Part I Finance 14 Visualization of the risk process Pawel Mista, Rafal Weron 14.1 Introduction 162 14 Visualization of the risk process 15 Approximation of ruin probability Krzysztof Burnecki, Pawel Mista, Aleksander Weron 15.1 Introduction 164 15 Approximation of ruin probability 16 Deductibles Krzysztof Burnecki, Joanna Nowicka-Zagrajek, Aleksander Weron, A Wyloma´ska n 16.1 Introduction 166 16 Deductibles 17 Premium calculation Krzysztof Burnecki, Joanna Nowicka-Zagrajek, W Otto 17.1 Introduction 168 17 Premium calculation 18 Premium calculation when independency and normality assumptions are relaxed W Otto 18.1 Introduction 17018 Premium calculation when independency and normality assumptions are relaxed 19 Joint decisions on premiums, capital invested in insurance company, rate of return on that capital and reinsurance W Otto 19.1 Introduction 17219 Joint decisions on premiums, capital invested in insurance company, rate of return on that ca 20 Stable Levy motion approximation in collective risk theory Hansjoerg Furrer, Zbigniew Michna, Aleksander Weron 20.1 Introduction 174 20 Stable Levy motion approximation in collective risk theory 21 Diffusion approximations in risk theory Zbigniew Michna 21.1 Introduction ... ranging between and 134 for different estimates of α and sample sizes Once α and σ have been obtained and α and σ have been fixed at these values, ˆ ˆ estimates of β and µ can be obtained using... of the American Statistical Association 71: 340–344 Embrechts, P., Kluppelberg, C and Mikosch, T (1997) Modelling Extremal Events for Insurance and Finance, Springer Fofack, H and Nolan, J P (1999)... lists various Archimedean copulae in the same ordering as in Table 2.1 in Hărdle, Kleinow and Stahl (2002) or in Nelsen (1999) and the corresponding a upper and lower TDC The quantlet TailCoeffCopula